a) Let X be a random variable with probability mass [k(x+4) if x=-2,-1, 0, 1 otherwise function(x)=0 i) the value of k ii) P(-1 i) Exactly three wells have the impurity assuming that the conjecture is correct? ii) Between 3 and 5 wells are impure? (2 Marks) (3 Marks) c) The proportion of the budget for a certain type of industrial company that is allotted to environmental and pollution control is a continuous random variable X with probability [s(1-x)' if 0 x I density function f(x)= {" otherwise i) Find the mean and variance of the proportion of budget allocated to environmental and pollution control. ii) What is the probability that a company chosen at random expends less than 10% of its (8 Marks) budget on environmental and pollution controls? d) Suppose that X is a random variable with moment generating function M, () = exp(1 +31). Find the moment generating function of the random variable Y=4-2X and hence identify the distribution of Y. (4 Marks) e) It has been observed that a certain measurement is a normal random variable of which 10% are less than 10.87 cm and 25% greater than 18.70 cm. What are the mean and standard deviation of the measurements? (4 Marks)